Three Well-Performing RLS Algorithms for Practical Implementation

In the following section, we present three well-performing RLS (Recursive Least Squares) algorithms for reference and practical implementation. These algorithms are designed to solve problems in domains such as linear filtering, signal processing, and machine learning. The first algorithm is a forward-recursive RLS implementation, which efficiently handles fitting problems for non-stationary signals. In code implementations, this typically involves initializing a covariance matrix and recursively updating weight vectors using a gain vector calculation, making it suitable for real-time adaptive filtering applications. The second algorithm utilizes backward recursion, offering computational advantages over forward recursion and delivering excellent performance with stationary signals. Implementation-wise, this approach often employs an inverse covariance matrix update with reduced computational complexity O(n²), frequently implemented using Python's NumPy for matrix operations or C++ for embedded systems. The final algorithm integrates Kalman filter theory with RLS, demonstrating exceptional performance in estimating state variables within linear systems. This hybrid implementation commonly features a state-space model representation where the RLS component handles parameter adaptation while Kalman filtering manages state estimation, often implemented using MATLAB's System Identification Toolbox or custom C++ libraries for high-performance applications. We believe these algorithms will prove highly valuable for professionals working in these domains, providing effective solutions to complex problems through optimized computational approaches and practical implementation frameworks.